Solution manual for an introduction to management science quantitative approaches to decision making

Page 1

Solution Manual for An Introduction to Management Science Quantitative Approaches to Decision Making

13th

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Chapter 1

Introduction

Learning Objectives

1. Develop a general understanding of the management science/operations research approach to decision making.

2. Realize that quantitative applications begin with a problem situation.

3. Obtain a brief introduction to quantitative techniques and their frequency of use in practice.

4. Understand that managerial problem situations have both quantitative and qualitative considerations that are important in the decision making process.

5. Learn about models in terms of what they are and why they are useful (the emphasis is on mathematical models).

6. Identify the step-by-step procedure that is used in most quantitative approaches to decision making.

7. Learn about basic models of cost, revenue, and profit and be able to compute the breakeven point.

8. Obtain an introduction to the use of computer software packages such as Microsoft Excel in applying quantitative methods to decision making.

9. Understand the following terms: model infeasible solution objective function management science constraint operations research deterministic model fixed cost stochastic model variable cost feasible solution breakeven point

1 - 1 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Solutions:

1. Management science and operations research, terms used almost interchangeably, are broad disciplines that employ scientific methodology in managerial decision making or problem solving. Drawing upon a variety of disciplines (behavioral, mathematical, etc.), management science and operations research combine quantitative and qualitative considerations in order to establish policies and decisions that are in the best interest of the organization.

2. Define the problem

Identify the alternatives

Determine the criteria

Evaluate the alternatives

Choose an alternative

For further discussion see section 1.3

3. See section 1.2.

4. A quantitative approach should be considered because the problem is large, complex, important, new and repetitive.

5. Models usually have time, cost, and risk advantages over experimenting with actual situations.

6. Model (a) may be quicker to formulate, easier to solve, and/or more easily understood.

7. Let d = distance

m = miles per gallon

c = cost per gallon, Total Cost

We must be willing to treat m and c as known and not subject to variation.

8. a. Maximize 10x + 5y

Chapter 1 1 - 2 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
2d c m   
=
s.t.
x + 2y  40 x  0, y  0
5

b. Controllable inputs: x and y

Uncontrollable inputs: profit (10,5), labor hours (5,2) and labor-hour availability (40)

c.

Profit:

$10/unit forx $ 5/ unit fory

Labor Hours: 5/unit forx 2/ unit fory

40 labor-hour capacity

d. x = 0, y = 20 Profit = $100 (Solution by trial-and-error)

e. Deterministic - all uncontrollable inputs are fixed and known.

9. If a = 3, x = 13 1/3 and profit = 133

If a = 4, x = 10 and profit = 100

If a = 5, x = 8 and profit = 80

If a = 6, x = 6 2/3 and profit = 67

Since a is unknown, the actual values of x and profit are not known with certainty.

10.

a. Total Units Received = x + y

b. Total Cost = 0.20x +0.25y

c. x + y = 5000

d. x  4000 Kansas City Constraint

y  3000 Minneapolis Constraint

Introduction 1 - 3 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Quantities x and y Controllable Input Projected Profit and check on production time constraint Output Max 10 x + 5 y s.t. 10 x y + 5 40  x y 0 0   Mathematical Model
Uncontrollable Inputs Production

e. Min 0.20x + 0.25y s.t. x + y = 5000 x  4000

3000

11. a. at $20 d = 800 - 10(20) = 600 at $70 d = 800 - 10(70) = 100

b. TR = dp = (800 - 10p)p = 800p - 10p2

c. at $30 TR = 800(30) - 10(30)2 = 15,000 at $40 TR = 800(40) - 10(40)2 = 16,000 at $50 TR = 800(50) - 10(50)2 = 15,000

Total Revenue is maximized at the $40 price.

d. d = 800 - 10(40) = 400 units TR = $16,000

12.

a. TC = 1000 + 30x

b. P = 40x - (1000 + 30x) = 10x - 1000

c. Breakeven point is the value of x when P = 0 Thus 10x - 1000 = 0 10x = 1000 x = 100

13. a. Total cost = 4800 + 60x

b. Total profit = total revenue - total cost = 300x - (4800 + 60x) = 240x - 4800

c. Total profit = 240(30) - 4800 = 2400

d. 240x - 4800 = 0

x = 4800/240 = 20

The breakeven point is 20 students.

14. a. Profit = Revenue - Cost = 20x - (80,000 + 3x) = 17x - 80,000

17x - 80,000 = 0 17x = 80,000

x = 4706

Breakeven point = 4706

Chapter 1 1 - 4
© 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
x
y 
, y  0

15.

b. Profit = 17(4000) - 80,000 = -12,000

Thus, a loss of $12,000 is anticipated.

c. Profit = px - (80,000 + 3x) = 4000p - (80,000 + 3(4000)) = 0 4000p = 92,000 p = 23

d. Profit = $25.95 (4000) - (80,000 + 3 (4000)) = $11,800

Probably go ahead with the project although the $11,800 is only a 12.8% return on the total cost of $92,000.

a. Profit = 100,000x - (1,500,000 + 50,000x) = 0 50,000x = 1,500,000 x = 30

b. Build the luxury boxes. Profit = 100,000 (50) - (1,500,000 + 50,000 (50)) = $1,000,000

16. a. Max 6x + 4y

b. 50x + 30y  80,000

Introduction 1 - 5 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
50
30
x, y  0 17. a. sj = sj - 1 + xj - dj or sj - sj-1 - xj +
xj  cj
sj  Ij
x  50,000
y  45,000
dj = 0 b.
c.

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